Hoo boy, this is a grind. I'll assume you're already Level 60 and have completed the main storyline quests for this guide.
Once you hit Level 60, several quests associated with the Elemental Evil questline will be available to you - these are your intended path from Level 60 to 70. Once you've defended the tree in Protector's Enclave, you'll begin the Vigilance portion. Moving through the following areas, you'll have to complete 3 sets of 8 vigilance tasks per area in order to complete the area (which I would recommend doing, these quests do provide gear that will be useful to you as you move toward L70). You'll want to complete these areas in this order Drowned Shore -> Reclamation Rock -> Fiery Pit -> Spinward Rise.
EDIT: As mentioned in the comments by wolfzero01, probably the easiest area to grind is Drowned Shore. The experience rewards are only slightly less than some of the other areas, and the enemies you encounter are significantly easier than in some of the higher areas. I'd still recommend spacing your time out across the areas, but this is a great suggestion if you find you're losing time dealing with higher level enemies and using injury kits (as I did).
My tips -
1) If you're not in a hurry to start seeing new content, just wait for one of the many Double XP weekends that this game has. We've already seen several, and once the content is a bit older, we'll almost certainly be seeing another experience multiplier (once Perfect World isn't making cash on their paid experience boosters :) ).
2) Related to the first point - keep an eye out for other promotions. From September 8 - 22, 2015, you could receive a free level's worth of experience (as well as a free artifact) by visiting the Claims Agent in Protector's Enclave (to the west of Lord Neverember's Chamber, opposite where Sergeant Knox posts up). Similar promotions will likely take place in the future, so keep an eye outEDIT - As Calex dEUS mentions in the comments, the free level's worth of experience will scale to provide you with enough experience to gain the current level you're working on. Thus, it would be most efficient to claim this for your last (or latest, if you're not 69 by September 22). Good advice if this comes back!
3) Make sure all of your utility slots are equipped with Azure Enchantments - these will increase the amount of experience awarded for your actions. Just slot whatever the most cost-efficient ranked enchantment you can afford is - I personally used Rank 6's, but obviously the higher the rank, the faster you'll level up.
4) Get a leveling buddy - many of the vigilance quests require you to kill X amount of enemies, and your party members' kills will also count in most cases (provided you're close enough when they make a kill). From my experience, this works best in smaller groups so that you can coordinate which quests and areas you're going to visit most effectively - I'd recommend groups of 2 or 3.
5) While it may seem tempting to level in the highest level area you can currently, most of the vigilance quests reward pretty similar amounts of experience. If you're having trouble with enemies in an area, try dropping back to an earlier area. Chances are that you'll get more experience just from moving through the quests faster.
6) Equipment - you'll probably want some decent equipment for the grind. My choice was the Eternal Armor, which can be purchased from the Tyranny of Dragons Quartermaster (in the same general area as the Claims Agent mentioned earlier). This equipment will level with you, and is a good choice if you've already got some of the campaign currency from Tyranny of Dragons. If not, don't fret - you'll find good equipment as you move through the new areas.
7) Premium Boosts - if you don't mind dropping real currency, you can buy experience boosts through the Zen Market. Your choice :)
7b) In addition to the experience boosts you can buy in the Zen Market, there is also Neverwinter VIP. A month can be purchased for 1000 Zen, and there are discounts for buying in bulk. Each month of VIP you purchase also provides you with added benefits (see https://www.arcgames.com/en/games/neverwinter/news/detail/94...
! for details). The rank one bonus is a 10% increase on all experience earned (and you get an enchanted key every day you log in). The general consensus is that this can usually pay for itself if you're logging in each day.
Anyway, regardless of how you approach it, this is going to be a grind. It takes more experience to go from 69 - 70 than it does to go from 1 - 60, so just take it at your own pace. If you've got any extra tips that I haven't mentioned that worked for you, please mention them in the comments, and let me know if there's anything I can do to improve this guide. Thanks, and good luck!